Properties

Degree 2
Conductor $ 7 \cdot 1429 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 4-s − 2·6-s − 7-s + 3·8-s + 9-s + 6·11-s − 2·12-s − 2·13-s + 14-s − 16-s − 4·17-s − 18-s − 4·19-s − 2·21-s − 6·22-s − 23-s + 6·24-s − 5·25-s + 2·26-s − 4·27-s + 28-s − 4·29-s − 3·31-s − 5·32-s + 12·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.436·21-s − 1.27·22-s − 0.208·23-s + 1.22·24-s − 25-s + 0.392·26-s − 0.769·27-s + 0.188·28-s − 0.742·29-s − 0.538·31-s − 0.883·32-s + 2.08·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10003\)    =    \(7 \cdot 1429\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{10003} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 10003,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.449939064\)
\(L(\frac12)\)  \(\approx\)  \(1.449939064\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;1429\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;1429\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 + T \)
1429 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.84541669874798, −16.19608955591976, −15.31212124501984, −14.88816147962213, −14.26210885965216, −13.92055202019061, −13.32579785923293, −12.70435724890919, −12.07293512323865, −11.19959504325112, −10.73707879332759, −9.676666008275327, −9.409779638767370, −9.078483341066316, −8.449385444682609, −7.845452314823444, −7.209972312904100, −6.487891661887569, −5.748865272651055, −4.603662742934374, −3.948899335510006, −3.624431856677182, −2.315390352799956, −1.851239306615050, −0.5927895354433112, 0.5927895354433112, 1.851239306615050, 2.315390352799956, 3.624431856677182, 3.948899335510006, 4.603662742934374, 5.748865272651055, 6.487891661887569, 7.209972312904100, 7.845452314823444, 8.449385444682609, 9.078483341066316, 9.409779638767370, 9.676666008275327, 10.73707879332759, 11.19959504325112, 12.07293512323865, 12.70435724890919, 13.32579785923293, 13.92055202019061, 14.26210885965216, 14.88816147962213, 15.31212124501984, 16.19608955591976, 16.84541669874798

Graph of the $Z$-function along the critical line